rela
tivity
Qu
an
tum
Classic physics
Modern physics
Einstein:The founder
of modern space-time
Time-space view
dynamics
Length contraction
Time
dilation
Relative nature of simultaneity
Prin
cip
le o
f P
rincip
le o
f re
lativ
ityre
lativ
ity
Lorentz transformatio
Lorentz transformatio
nn
Relativity
Einstain’s theory took us into a world far beyond that of ordinary experience,it led us to a deeper and more satisfying view of the nature of space and time
In modern long range navigation,the precise location and speed of moving craft are continuously monitored and updated.A system of navigation satellites called NAVSTAR permits locations and speeds anywhere on earth to be determined to within about 16m and 2cm/s.However if relativity effects were not taken into account,speeds could not be determined any closer than about 20cm/s,which is unacceptable for modern navigation systems.How can something as abstract as Einstain’s relativity be involved in something as practical as navigation?
A pair of twins ,A remains on earth,and B make the milk run to a nearby solar system in high speed,when B come back,who’s younger?
Take an account on a event in two different frame
1.Galileo transformation
Pro
Su
xxo
S
r
§1 Galileo relativity principle§1 Galileo relativity principle
utxx yy
zz tt
tt
zz
yy
tuxx
S
S
tzyxr ,,,
tzyxr ,,,
tzyxv ,,,
tzyxv ,,,
a
a
Pro
Su
xxo
S
r
Velocity and acceleration transformation
zz
yy
xx
vv
vv
uvv
zz
yy
xx
vv
vv
uvv
zz
yy
xx
aa
aadt
duaa
zz
yy
xx
aa
aatd
duaa
zz
yy
xx
aa
aa
aa
zz
yy
xx
aa
aa
aa
u
is constant
In two inertial frame aa
conclusion: 1.time interval is absolute
t=t t=t2.space separation is absolute
tuxx t u x t u x x
0t xx 3.the invariability of Newton’s law
ammaF
for : so :
amF FF
P957
S 2021012211 vmvmvmvm
S 2021012211 vmvmvmvm
For example : conservation of momentum
2.Galileo relativity
3.the trouble in electromagnetic equation1)
01
2
2
22
2
t
E
cx
E
C :to which reference frame ?
Even though electromagnetism shares with mechanics concept such as energy and momentum.there appear to be a major difference between these two fundamental discipline.the laws of mechanics look the same in all inertial frames,but electromagnetism appears to violate the general law.According to Maxwell’s equation,electromagnetic waves propagate at speed C,with no restrictions on the state of the source of detector,this suggests the existence of an absolute frame for electromagnetism.
1.the relativity postulate:the laws of physics are the same for observers in all inertial reference frames.no frames is preferred
2.the speed postulate:the speed of light in vacuum has the same value c in all direction and in all inertial reference frames
Einstein relativity develop Newton’s theory
discussion
physicsrule
Mechanics rule
§2 the postulates
The light speed invariability is opposed to Galileo velocity transformation Difference in view
Newton
Time scale
Length scale
Mass measure
Has no relation
with frame
relativity Time,space,mass has relation with reference frame
C is
constant
transformation Inverse transformation
xc
tt
zz
yy
utxx
xc
tt
zz
yy
tuxx
2.lorentz transformation1.transformation formula
21
1
c
u
ttux ,,
1
tt
zz
yy
utxx
Galileo
transformation
cu <<
discussionHas relation with
No meaning.Maximum Speed is C
cu >
4.procedure to solve the problem
1)establish coordinate
2)determine the moving frame and rest frame
3)u is the positive speed of S’ to S
4)use formula to get relation among x, t, x’
t’ and solve problem
P
o
Su
xxo
S
Example:two persons a,b observe lighting pulses, from point of a, x1=6104m , t1=2 10-4 s ; x2=12 104m , t2=1 10-4 s , from the point of b, two events happen at the same time ( 1 ) find the relative velocity of b to a ( 2 ) find the space separation of lighting pulses measured by b
x
c
vtt 221
1
2
2
1
xc
vt
t
Solution:
We get
0t
2
2
44
2
44
1
)1061012()102101(0
c
vc
v
2
cv
From lorentz transformation
vtxx
21
1
mtvx
x 4
21020.5
1
Example:a race track with length 100m,a sports man
Run from the origin point to end point with time interval 10s,a craft with velocity 0.8c fly along the direction of run way.from the point of craft man
find the space separation and time interval
Solution:x=100, t=10s,u=0.8c
mtux
x 9
2104
1
Negative sign means the sportsman run in the opposite direction
sx
cu
tt 6.16
1 2
2
§ 2 time and space of relativity§ 2 time and space of relativity1 、 relative nature of simultaneity
But in Einstain theory:
00 ttIn Newton’s theory
0
0
0
t
x
t
t
tv
xcv
c
2
2
21
?tso,
Relative nature of simultaneity 。
?0 tt
if
We have
Conclusion:in general,two events that appear simultaneous in one frame of reference don’t appear simultaneous in a second,unless the two events happen in the same place.
Einstein trainSS
S rest frame
In train
There’s a signal source
0tt M Give a signal
In the middle
M
S uA BM
Einstain’s train experiment
Event 1
Receive flashA
Event 2 Receive flashB
S MBMA
A B Receive the flash at same time
S Su
A BM
Two events happen in different time
M Give a signal BA Move with
S
Receive light early than
A
0tt M Give a signal
BS
discussion
S Su
A BM
1 ) simultaneous is absolutely only when two events happen in same place
tt
vx
cv
c
2
2
21
2) Relative nature of simultaneous is the result of constant c
3) When speed is far more less than C,the result in two inertial frame is same
tt
vx
cv
c
2
2
21
A君
B君
Example:two trains leave from two station A,B with space separation 1000km at the same time,a craft with u=9km/s along the direction ab,find
The interval from the point of spacemanSolution:x=106 t=0
7
2
2
101
xcu
tt
Negative means b train go first
1)proper time
a time interval between two events at the same space point in a frame is called a proper time in that frame
2. Time dilation
2 、 proper time is shortest time in all frames Take an account on a clock in S
0x t
2
2
2
1cu
xcu
tt
2
2
1cu
t
t
x 0
1 t
Time dilation,proper time is minimum time
In s,from lorentz transformation
3)physics reason of time dilation y′
x′
u
d
u t
dl
M′
A′C′ C′
In S ′, there’s a light source in A ′ M′is a reflected mirror
c
dtx
20
y′
x′
u
d
u t
dl
M′
A′C′ C′
In S :
22
2
22
0
tu
dcc
lt
x
2
21
2
cu
cd
t
2
21
cu
tt
tt
Time dilation
(2)the proper time is the shortest one in all frames
discussion
( 1 ) time dilation is the space-time effect of relativity,it has no relation with the structure of clock
( 3 ) there’re a lot of experiment to proof the time dilation effect
example a rocket v=0.95c , the time interval is 10min measured in rocket , how long is it in earth frame ?
min01.32min95.01
10
1 22
'
tt
Solution:
Example:the lifelong time for is 2.5×10-8s while in rest,if its u=0.99c,the passing distance is 52m,is this ok ?
Solution:if with t′=2.5 ×10-8s times u , we get 7.4m 。Take account of time dilation
2
2
1cu
tt
)(108.1
)99.0(1
105.2 7
2
8
s
=
So s=uΔt=53m,it’s ok
3.length contraction S
uS
0l1 、 proper length
A length measured in the rest frame of the body is called proper length
2 、 proper length is the longest in all frames (length contraction)
S
S 120 xxl
12 xxl
012 ttt
Notes:in S we measure the rod length,we must measure end points in same time
2
2
1cu
tuxx
From lorenze transformation
2
2
0 1c
ull
2
2
0 1c
ull 11 0ll
Proper length is longest
discussiondiscussion
1)Relativity effect
2) In low speed Galileo transformation
4)length contraction is relativity effect,it’s different from what we say that the body become smaller.
3)proper length is the longest ,0ll
Example:a length of rocket measured in rocket frame is 15m, suppose v=0.95c,find the length in earth frame ?
2
0 1 ll
mml 6.495.0115 2
Solution:
cv2
3
Example:a 1m rod rest in O’x’y’ 。 the angle is 450 with x’ axis measured from s’ 。 find the length of the rod and angle with x in s 。 The related velocity is
'''''' sincos llll yx
solution:
z
y y'
S S'v
O O
z'
x
x'
ly'
lx''
l'
from lorentz transformation:
''' sinlll yy
'22'22 cos1 llll yx
2' 1cos llx
2
'
'2'
''
1cos1
sin
tg
l
l
l
ltg
x
y
,23
,1,45 '0' cvml
0
2
'
43.63,21
tg
tg
mll 79.0cos1 '22'
Conclusion:not only the rod have contraction,but also rotate in some angle.
Example:in 6000m altitude , a meson fly to earth with v=0.998c 。 Suppose meson ‘s life span in its rest frame is 210-6 s , from relativity , 1)can meson arrive in earth from earth frame ? 2) from mesonFrame?
s
cv
tt 6
2
0 106.31
1
solution : 1) proper time for meson is t0=210-6 s 。 Because of time dilation , the life span from earth
The distance for to travel m6000m9460 tvL
It can go straight through the earth
2 ) from frame
mc
c
c
ull 360)
998.0(160001 2
2
2
0
The distance for to travel in frame
mcvl 599102998.0 6
It can go straight through the earth
4 、 comparison of two time-space view
space,time is absolutely,there’r no relation among time,space and the motion of body
Classic view
4 、 light speed is c,which is utmost speed of motion body
Relativity view
1 、 there are relation among time space and the motion of body
2 、 every inertial frame has its own time scale,and found that the clock in other frame go slow
3 、 every inertial frame has its own space scale,and found the ruler in other frame become shorter.
§ 5 the dynamic of relativity§ 5 the dynamic of relativity 1 、 momentum and dynamic equation
vmp
m
But the utmost of v is c
2
2
0
1cv
mm
dt
Must change with speed
0
dtmF
v
mF
dtdv
vmp A:It can be
proofed
notes :
m m 0( 1 ) if , v c
( 2 ) if , cv m
( 3 ) if , v c 00m
B: dynamic equation
vmp
d
dtv F
m
v c( )0
2 21
v cif amF
0
(4) v>c , m is negative,no meaning
2 、 mass and energy
1 、 kinetic energy
2
0
2 cmmcEK
If v<<c:
2
0
2
2
2
0k 21 E cmc
c
vm
2
02
1 vm
2
2
0
1cv
mm
2
2
2
2 21
11
1cv
cv
p986
三、相对论能量 质能关系
2
0
2 cmmcEK
Rest energyKinetic energy Total energy
Mass and energy formula
56
2 、 energy
Notes:
( 1 ) a body in rest,it still has substantial
energy 。
( 2 ) mass is not the measure of inertial,but also the energy
( 3 ) change in mass implies change in energy
( 4 ) to isolated system,the total energy keeps constant
In 1955,atomic age has arrive!
application
Example:there’re mass loss because of radiation energy of the sun
S
ESEr
sJt
E/1029.4 26
skgtc
E
t
m/104.5 9
2
stm
m/105.8 14
3 、 the relation between momentum and energy
In relativity p mvvm
v c
0
2 21
E mcm c
v c
2 0
2
2 21From above
2 2 2 2 220
2( ) ( )mc m c m v c
2 2 2 20E E p c
E0E
pc
0E
kE
Basic formula in dynamics
2 2 2 20E E p c
2
0
2 cmmcEK
2
2
0
1cv
mm
Example:a particle move with v=0.80c 。 Find the total energy,kinetic energy and momentum
MeV
MeV
cv
cmmcE
1563
8.01(
938
)/1(
2/12
2/122
2
02
)
Kinetic energy
MeVMeVMeVcmEEk 62593815632
0
Solution: E0=m0c2=938MeV
momentum
119
1
2/12
827
2/122
0
..1068.6
..8.01(
1038.01067.1
)/1(
smkg
smkg
cv
cmmvp
)
Alternative way for momentum
MeVMeVcmEcp 12509381563)( 2222
0
2
cMeVp /1250
Example: m0 of electron is known ( 1 ) find the E0 ;( 2 ) find the work for the particle moving from rest to 0.60c
2 ) the work for accelerating particle
J10025.160.01
10199.8
1
13
2
14
2
22
cmmcE o
J 10 05 . 214
k E
evcmE 52
0 1012.50
solution :( 1 ) rest energy of electronic
0EEE
k
Example: two rest particle with rest mass m0 each collide head on with v to become a combined particle 。 Find the rest mass and speed of the combined particle 。
2
202
1
2
cmMc
MVvmvm -
2
0
1
2
mM
Solution from the conservation law of momentum and energy
Notes:the rest mass of combined particle is larger than 2m0 , the difference
2
2)
1(
20
221
02
02
02
02
2
0
2
c
KE
cmcm
cm
mmM
--
The difference of rest mass comes from kinetic energy
V=0
(1)
(2)
We’v